<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> &amp; ( for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">VV</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">VV</font> &amp; ( for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">VV</font> holds <br/><font color="Olive" title="b3">W</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a> ) ) ) implies  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E31">A1</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b2">X</font> &amp; ( for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Olive" title="b3">x</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being   <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b2">x</font></span>)</span> where <font color="Olive" title="b2">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span> </span><br/>

<b>set </b><font color="Maroon" title="c2">IPt</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c3">IPs</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<a NAME="E3:50"/><i><font color="Green" title="E33">A3</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T23" target="_self" title="WAYBEL14:th.23">Th23</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E34">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E35">A5</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T23" target="_self" title="WAYBEL14:th.23">Th23</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span> is <a href="waybel_8.html#V1" title="WAYBEL_8:attr.1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="waybel_8.html#D4" title="WAYBEL_8:def.4">WAYBEL_8:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">ips</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span> as  <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel14.html#E1:50"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="waybel14.html#E5:50"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c4">ips</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
;<br/>
<a NAME="E9:50"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  ex <font color="Olive" title="b2">X</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b1">x</font> st <br/>for <font color="Olive" title="b3">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/>( <font color="Olive" title="b3">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:50"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="waybel14.html#T39" target="_self" title="WAYBEL14:th.39">Th39</a></i>;<br/>
<a NAME="E10:50"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b2">V</font></span>)</span> where <font color="Olive" title="b2">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font> ) </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E8:50"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="waybel14.html#T37" target="_self" title="WAYBEL14:th.37">Th37</a></i>;<br/>
<a NAME="E11:50"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T38" target="_self" title="WAYBEL14:th.38">Th38</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span> ;<br/>
<a NAME="E12:50"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:50_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">B</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E15:50"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:50_2"/><i><font color="Green" title="E40">A10</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> )
 ;<br/>
<a NAME="E4:50_2"/>
<font color="Maroon" title="c9">k</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E5:50_2"/><b>then </b>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c9">k</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T2" title="WAYBEL_8:th.2">WAYBEL_8:2</a></i>;<br/>
<a NAME="E6:50_2"/><b>then </b>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c9">k</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T41" title="WAYBEL11:th.41">WAYBEL11:41</a></i>;<br/>
<b>hence </b><a NAME="E7:50_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Bx</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c8">x</font> st <font color="Olive" title="b1">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><b>set </b><font color="Maroon" title="c9">Bx</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> ) </span>}</span> ;<br/><a NAME="E1:50_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> ) </span>}</span>  or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:50_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c10">Bx</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : (  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>reconsider </b><font color="Maroon" title="c11">Bx</font> = <font color="Maroon" title="c10">Bx</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E4:50_3"/>
<font color="Maroon" title="c11">Bx</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c8">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_3_2"/>
<font color="Maroon" title="c11">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:50_3_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_2_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E15:50"><i><font color="Green" title="E39">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_3_2_2_1"/>
( <font color="Maroon" title="c11">Bx</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c11">Bx</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:50_3_2_2_1_1"/>
<font color="Maroon" title="c11">Bx</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font></span><br/><div class="add"><a NAME="E2:50_3_2_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:50_3_2_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:50_3_2_2_1_2"/><i><font color="Green" title="E40">A11</font></i>: 
not <font color="Maroon" title="c11">Bx</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font></span><br/><div class="add"><a NAME="E2:50_3_2_2_1_2"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c11">Bx</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font> )</span><br/><b>assume </b><a NAME="E1:50_3_2_2_1_2_1"/>
<font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font></span><br/><a NAME="E2:50_3_2_2_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c12">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> )
 ;<br/><b>hence </b><a NAME="E3:50_3_2_2_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_3_2_2_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_2_1_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="waybel14.html#E2:50_3_2_2_1_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_3_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">Bx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c12">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">S</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:50_3_2"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c12">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:50_3_2"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c13">S'</font> = <font color="Maroon" title="c12">S</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="waybel14.html#E4:50_3_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:50_3_2"/>
<font color="Maroon" title="c13">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c13">S'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E6:50"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="waybel_8.html#D3" title="WAYBEL_8:def.3">WAYBEL_8:def 3</a></i>;<br/>
<a NAME="E8:50_3_2"/><b>then </b>
<font color="Maroon" title="c13">S'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c13">S'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">UA</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:50_3_2"/><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">UA</font> &amp; <font color="Maroon" title="c14">UA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_8.html#K2" title="WAYBEL_8:func.2">compactbelow</a> <font color="Maroon" title="c13">S'</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E5:50_3_2"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">UA</font> = <font color="Maroon" title="c14">UA</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/>
<a NAME="E12:50_3_2"/><i><font color="Green" title="E43">A16</font></i>: 
( <font color="Maroon" title="c15">UA</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">S'</font> &amp; <font color="Maroon" title="c15">UA</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="waybel_8.html#T4" title="WAYBEL_8:th.4">WAYBEL_8:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">F</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:50_3_2"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c16">F</font>
 <b>and </b><br/><a NAME="E15:50_3_2"/><i><font color="Green" title="E45">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">F</font> holds <br/><font color="Olive" title="b1">x</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:50"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/>
<a NAME="E16:50_3_2"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E14:50_3_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>;<br/>
<a NAME="E17:50_3_2"/>
<font color="Maroon" title="c15">UA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">UA'</font> = <font color="Maroon" title="c15">UA</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E19:50_3_2"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c15">UA</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:50_3_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="waybel_3.html#D2" title="WAYBEL_3:def.2">WAYBEL_3:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">F'</font> = <font color="Maroon" title="c16">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E21:50_3_2"/>
<font color="Maroon" title="c18">F'</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">F'</font> or <font color="Maroon" title="c19">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_3"/>
<font color="Maroon" title="c19">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">F'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>hence </b><a NAME="E2:50_3_2_3"/>
<font color="Maroon" title="c19">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c19">G</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c18">F'</font><b> such that </b><br/><a NAME="E23:50_3_2"/><i><font color="Green" title="E48">A21</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c19">G</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="waybel14.html#E19:50_3_2"><i><font color="Green" title="E47">A20</font></i></a>, <a class="ref" href="waybel_3.html#T34" title="WAYBEL_3:th.34">WAYBEL_3:34</a></i>;<br/>
<a NAME="E24:50_3_2"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c19">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E25:50_3_2"/><b>then </b><i><font color="Green" title="E49">A22</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c19">G</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="waybel14.html#E23:50_3_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">G</font> = <font color="Maroon" title="c19">G</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">Gg</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E28:50_3_2"/><i><font color="Green" title="E50">A23</font></i>: 
<font color="Maroon" title="c21">Gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">G</font>
 <b>and </b><br/><a NAME="E29:50_3_2"/><i><font color="Green" title="E51">A24</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c21">Gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c20">G</font>
 <b>and </b><br/><a NAME="E30:50_3_2"/><i><font color="Green" title="E52">A25</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Gg</font> holds <br/> not <font color="Olive" title="b1">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel14.html#T1" target="_self" title="WAYBEL14:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">U1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E32:50_3_2"/><i><font color="Green" title="E53">A26</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">U1</font> &amp; <font color="Maroon" title="c22">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Gg</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="waybel14.html#E23:50_3_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="waybel14.html#E29:50_3_2"><i><font color="Green" title="E51">A24</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E33:50_3_2"/><i><font color="Green" title="E54">A27</font></i>: 
<font color="Maroon" title="c21">Gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E28:50_3_2"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E34:50_3_2"/><b>then </b>
<font color="Maroon" title="c22">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">F</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">U1</font> = <font color="Maroon" title="c22">U1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E36:50_3_2"/>
<font color="Maroon" title="c23">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">U1'</font> = <font color="Maroon" title="c23">U1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E38:50_3_2"/>
<font color="Maroon" title="c23">U1</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E15:50_3_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="waybel14.html#E33:50_3_2"><i><font color="Green" title="E54">A27</font></i></a></i>;<br/>
<a NAME="E39:50_3_2"/><b>then </b><i><font color="Green" title="E55">A28</font></i>: 
( <font color="Maroon" title="c24">U1'</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> &amp; <font color="Maroon" title="c24">U1'</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 
<b>by </b><i><a class="ref" href="waybel14.html#T27" target="_self" title="WAYBEL14:th.27">Th27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">k</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4"/><i><font color="Green" title="E56">A29</font></i>: 
not  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>set </b><font color="Maroon" title="c26">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c27">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_3_2_4"/><i><font color="Green" title="E57">A30</font></i>: 
<font color="Maroon" title="c27">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">u</font> = <font color="Maroon" title="c27">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4"><i><font color="Green" title="E57">A30</font></i></a></i>;<br/><a NAME="E5:50_3_2_4"/><i><font color="Green" title="E58">A31</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c28">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4"><i><font color="Green" title="E57">A30</font></i></a></i>;<br/><a NAME="E6:50_3_2_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span>  or <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_4_1"/>
<font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c30">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:50_3_2_4_1"/><i><font color="Green" title="E59">A32</font></i>: 
( <font color="Maroon" title="c29">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c30">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c30">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> )
 ;<br/>
<a NAME="E4:50_3_2_4_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c30">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T12" title="WAYBEL11:th.12">WAYBEL11:12</a></i>;<br/>
<b>hence </b><a NAME="E5:50_3_2_4_1"/>
<font color="Maroon" title="c29">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_1"><i><font color="Green" title="E59">A32</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c29">D</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E5:50_3_2_4"><i><font color="Green" title="E58">A31</font></i></a>, <a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/><a NAME="E8:50_3_2_4"/><i><font color="Green" title="E59">A33</font></i>: 
<font color="Maroon" title="c29">D</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E39:50_3_2"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="waybel14.html#T25" target="_self" title="WAYBEL14:th.25">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4_2"/>
not <font color="Maroon" title="c15">UA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c29">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c30">l</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:50_3_2_4_2"/><i><font color="Green" title="E60">A34</font></i>: 
( <font color="Maroon" title="c30">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">UA'</font> &amp; not <font color="Maroon" title="c30">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c29">D</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">l</font> = <font color="Maroon" title="c30">l</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green" title="E60">A34</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c32">Uk</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:50_3_2_4_2"/><i><font color="Green" title="E61">A35</font></i>: 
( <font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c32">Uk</font> &amp; <font color="Maroon" title="c32">Uk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Gg</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E23:50_3_2"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="waybel14.html#E29:50_3_2"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E7:50_3_2_4_2"/><i><font color="Green" title="E62">A36</font></i>: 
<font color="Maroon" title="c21">Gg</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">F</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E28:50_3_2"><i><font color="Green" title="E50">A23</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:50_3_2_4_2"/><b>then </b>
<font color="Maroon" title="c32">Uk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">F</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green" title="E61">A35</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c33">Uk</font> = <font color="Maroon" title="c32">Uk</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ;<br/><a NAME="E10:50_3_2_4_2"/>
<font color="Maroon" title="c33">Uk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c34">Uk'</font> = <font color="Maroon" title="c33">Uk</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E12:50_3_2_4_2"/>
<font color="Maroon" title="c33">Uk</font> is <a href="waybel_6.html#V6" title="WAYBEL_6:attr.6">co-prime</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E15:50_3_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green" title="E61">A35</font></i></a>, <a class="txt" href="waybel14.html#E7:50_3_2_4_2"><i><font color="Green" title="E62">A36</font></i></a></i>;<br/><a NAME="E13:50_3_2_4_2"/><b>then </b><i><font color="Green" title="E63">A37</font></i>: 
( <font color="Maroon" title="c34">Uk'</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> &amp; <font color="Maroon" title="c34">Uk'</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> )
 <b>by </b><i><a class="ref" href="waybel14.html#T27" target="_self" title="WAYBEL14:th.27">Th27</a></i>;<br/><a NAME="E14:50_3_2_4_2"/><i><font color="Green" title="E64">A38</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c24">U1'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4_2_1"/>
not <font color="Maroon" title="c31">l</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c24">U1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c35">m</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:50_3_2_4_2_1"/><i><font color="Green" title="E65">A39</font></i>: 
( <font color="Maroon" title="c35">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> &amp; not <font color="Maroon" title="c31">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c35">m</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/><a NAME="E4:50_3_2_4_2_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c35">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">D</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2_1"><i><font color="Green" title="E65">A39</font></i></a></i>;<br/><a NAME="E5:50_3_2_4_2_1"/><b>then </b>
not <font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c35">m</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E6:50_3_2_4_2_1"/><b>then </b>
<font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c35">m</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E7:50_3_2_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2_1"><i><font color="Green" title="E65">A39</font></i></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E16:50_3_2_4_2"/><b>then </b>
<font color="Maroon" title="c31">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/><a NAME="E17:50_3_2_4_2"/><b>then </b><i><font color="Green" title="E65">A40</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">Uk'</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green" title="E61">A35</font></i></a>, <a class="txt" href="waybel14.html#E13:50_3_2_4_2"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/><a NAME="E18:50_3_2_4_2"/><i><font color="Green" title="E66">A41</font></i>: 
<font color="Maroon" title="c24">U1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">Uk</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> or <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Uk</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_4_2_2"/><i><font color="Green" title="E67">A42</font></i>: 
<font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Uk</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c36">d</font> = <font color="Maroon" title="c35">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:50_3_2_4_2_2"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c36">d</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E14:50_3_2_4_2"><i><font color="Green" title="E64">A38</font></i></a>, <a class="txt" href="waybel14.html#E1:50_3_2_4_2_2"><i><font color="Green" title="E67">A42</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:50_3_2_4_2_2"/>
<font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">Uk</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E13:50_3_2_4_2"><i><font color="Green" title="E63">A37</font></i></a>, <a class="txt" href="waybel14.html#E17:50_3_2_4_2"><i><font color="Green" title="E65">A40</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E19:50_3_2_4_2"/>
<font color="Maroon" title="c24">U1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">U1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c35">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> or <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">U1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_4_2_3"/><i><font color="Green" title="E67">A43</font></i>: 
<font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">U1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span><br/>

<a NAME="E2:50_3_2_4_2_3"/>
<font color="Maroon" title="c33">Uk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">U1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_4"><i><font color="Green" title="E56">A29</font></i></a>, <a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green" title="E61">A35</font></i></a>, <a class="txt" href="waybel14.html#E17:50_3_2_4_2"><i><font color="Green" title="E65">A40</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E3:50_3_2_4_2_3"/>
<font color="Maroon" title="c35">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">Gg</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c24">U1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel14.html#E18:50_3_2_4_2"><i><font color="Green" title="E66">A41</font></i></a>, <a class="txt" href="waybel14.html#E1:50_3_2_4_2_3"><i><font color="Green" title="E67">A43</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:50_3_2_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E30:50_3_2"><i><font color="Green" title="E52">A25</font></i></a>, <a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E10:50_3_2_4"/><b>then </b>
<font color="Maroon" title="c15">UA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c29">D</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T22" title="YELLOW_1:th.22">YELLOW_1:22</a></i>;<br/><a NAME="E11:50_3_2_4"/><b>then </b>
<font color="Maroon" title="c15">UA</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c29">D</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font></span>)</span><b> such that </b><br/><a NAME="E13:50_3_2_4"/><i><font color="Green" title="E60">A44</font></i>: 
<font color="Maroon" title="c30">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c29">D</font>
 <b>and </b><br/><a NAME="E14:50_3_2_4"/><i><font color="Green" title="E61">A45</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c30">d</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E19:50_3_2"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="waybel14.html#E8:50_3_2_4"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="waybel_3.html#D1" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c31">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E16:50_3_2_4"/><i><font color="Green" title="E62">A46</font></i>: 
( <font color="Maroon" title="c30">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c31">u</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c31">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E13:50_3_2_4"><i><font color="Green" title="E60">A44</font></i></a></i>;<br/><a NAME="E17:50_3_2_4"/>
<font color="Maroon" title="c23">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">UA</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="waybel14.html#E33:50_3_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E18:50_3_2_4"/><b>then </b><i><font color="Green" title="E63">A47</font></i>: 
<font color="Maroon" title="c31">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">UA</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2_4"><i><font color="Green" title="E62">A46</font></i></a></i>;<br/><a NAME="E19:50_3_2_4"/>
<font color="Maroon" title="c15">UA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c30">d</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E14:50_3_2_4"><i><font color="Green" title="E61">A45</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/><a NAME="E20:50_3_2_4"/><b>then </b><i><font color="Green" title="E64">A48</font></i>: 
not <font color="Maroon" title="c31">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c31">u</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2_4"><i><font color="Green" title="E62">A46</font></i></a>, <a class="txt" href="waybel14.html#E18:50_3_2_4"><i><font color="Green" title="E63">A47</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E21:50_3_2_4"/>
<font color="Maroon" title="c31">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c31">u</font>
 ;<br/><b>hence </b><a NAME="E22:50_3_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E20:50_3_2_4"><i><font color="Green" title="E64">A48</font></i></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E41:50_3_2"/><b>then </b><i><font color="Green" title="E56">A49</font></i>: 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c24">U1'</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E39:50_3_2"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="waybel11.html#T42" title="WAYBEL11:th.42">WAYBEL11:42</a></i>;<br/>
<a NAME="E42:50_3_2"/><i><font color="Green" title="E57">A50</font></i>: 
<font color="Maroon" title="c24">U1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:50_3_2_5"/><i><font color="Green" title="E58">A51</font></i>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">U1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c27">x'</font> = <font color="Maroon" title="c26">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:50_3_2_5"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c24">U1'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33" title="YELLOW_0:th.33">YELLOW_0:33</a></i>;<br/>
<a NAME="E4:50_3_2_5"/><b>then </b>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c27">x'</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_5"><i><font color="Green" title="E58">A51</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:50_3_2_5"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E43:50_3_2"/><b>then </b><i><font color="Green" title="E58">A52</font></i>: 
<font color="Maroon" title="c24">U1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E41:50_3_2"><i><font color="Green" title="E56">A49</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E44:50_3_2"/>
<font color="Maroon" title="c24">U1'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E45:50_3_2"/><b>then </b>
<font color="Maroon" title="c24">U1'</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:50"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="waybel14.html#E39:50_3_2"><i><font color="Green" title="E55">A28</font></i></a>, <a class="ref" href="waybel11.html#T46" title="WAYBEL11:th.46">WAYBEL11:46</a></i>;<br/>
<a NAME="E46:50_3_2"/><b>then </b><i><font color="Green" title="E59">A53</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> is <a href="waybel_3.html#V1" title="WAYBEL_3:attr.1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E43:50_3_2"><i><font color="Green" title="E58">A52</font></i></a>, <a class="ref" href="waybel_8.html#T2" title="WAYBEL_8:th.2">WAYBEL_8:2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c26">V</font> =  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font> &amp; <font color="Maroon" title="c26">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font> )</span><br/>

<a NAME="E47:50_3_2"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E46:50_3_2"><i><font color="Green" title="E59">A53</font></i></a>, <a class="ref" href="waybel_8.html#D1" title="WAYBEL_8:def.1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E48:50_3_2"/><b>then </b>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c24">U1'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 
;<br/>
<b>hence </b><a NAME="E49:50_3_2"/>
<font color="Maroon" title="c26">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Bx</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="txt" href="waybel14.html#E42:50_3_2"><i><font color="Green" title="E57">A50</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c26">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font></span><br/>

<a NAME="E50:50_3_2"/><i><font color="Green" title="E60">A54</font></i>: 
<font color="Maroon" title="c15">UA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:50_3_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<a NAME="E51:50_3_2"/>
<font color="Maroon" title="c23">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">UA</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E25:50_3_2"><i><font color="Green" title="E49">A22</font></i></a>, <a class="txt" href="waybel14.html#E29:50_3_2"><i><font color="Green" title="E51">A24</font></i></a>, <a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green" title="E53">A26</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E52:50_3_2"/>
<font color="Maroon" title="c26">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">S</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E43:50_3_2"><i><font color="Green" title="E58">A52</font></i></a>, <a class="txt" href="waybel14.html#E50:50_3_2"><i><font color="Green" title="E60">A54</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c12">Bx</font> = <font color="Maroon" title="c11">Bx</font> as    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c8">x</font> ;<br/><b>take </b><font color="Maroon" title="c13">Bx</font> = <font color="Maroon" title="c12">Bx</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font></span><br/><b>thus </b><a NAME="E6:50_3"/>
<font color="Maroon" title="c13">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Bx</font> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> )</span><br/>

<b>assume </b><a NAME="E1:50_3_3"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Bx</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font></span><br/>

<a NAME="E2:50_3_3"/><b>then </b>
 ex <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">k</font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_3"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">B</font> = <font color="Maroon" title="c7">B</font> as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E15:50"><i><font color="Green" title="E39">A9</font></i></a>, <a class="ref" href="yellow_8.html#T23" title="YELLOW_8:th.23">YELLOW_8:23</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">B</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span> </span><br/>

<b>thus </b><a NAME="E18:50"/>
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1" title="WAYBEL_8:func.1">CompactSublatt</a> <font color="Maroon" title="c1">L</font></span>)</span> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
